Quantale-valued Cauchy tower spaces and completeness

Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.

Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces

The operators D C Φ and C Φ D are defined by D C Φ f = f ∘ Φ ′ and C Φ D f = f ′ ∘ Φ where Φ is an analytic self-map of the unit disc and f is analytic in the disc. A characterization is provided for boundedness and compactness of the products of composition and differentiation from the spaces of fractional Cauchy transforms F α to the Bloch-type spaces B β , where α > 0 and β > 0 . In the case β < 2 , the operator D C Φ : F α ⟶ B β is compact ⇔ D C Φ : F α ⟶ B β is bounded ⇔ Φ ′ ∈ B β , Φ Φ ′ ∈ B β and Φ ∞ < 1 . For β < 1 , C Φ D : F α ⟶ B β is compact ⇔ C Φ D : F α ⟶ B β is bounded ⇔ Φ ∈ B β and Φ ∞ < 1 .

Weighted Composition Operators between the Fractional Cauchy Spaces and the Bloch-Type Spaces

We characterize boundedness and compactness of weighted composition operators mapping the families of fractional Cauchy transforms into the Bloch-type spaces. Corollaries are obtained about composition operators and multiplication operators.